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- Adaptive mesh refinement (2) (remove)
In (dynamic) adaptive mesh refinement (AMR) an input mesh is refined or coarsened to the need of the numerical application. This refinement happens with no respect to the originally meshed domain and is therefore limited to the geometrical accuracy of the original input mesh. We presented a novel approach to equip this input mesh with additional geometry information, to allow refinement and high-order cells based on the geometry of the original domain. We already showed a limited implementation of this algorithm. Now we evaluate this prototype with a numerical application and we prove its influence on the accuracy of certain numerical results. To be as practical as possible, we implement the ability to import meshes generated by Gmsh and equip them with the needed geometry information. Furthermore, we improve the mapping algorithm, which maps the geometry information of the boundary of a cell into the cell's volume. With these preliminary steps done, we use out new approach in a simulation of the advection of a concentration along the boundary of a sphere shell and past the boundary of a rotating cylinder. We evaluate the accuracy of our approach in comparison to the conventional refinement of cells to answer our research question: How does the performance and accuracy of the hexahedral curved domain AMR algorithm compare to linear AMR when solving the advection equation with the linear finite volume method? To answer this question, we show the influence of curved AMR on our simulation results and see, that it is even able to outperform far finer linear meshes in terms of accuracy. We also see that the current implementation of this approach is too slow for practical usage. We can therefore prove the benefits of curved AMR in certain, geometry-related application scenarios and show possible improvements to make it more feasible and practical in the future.
In tree-based adaptive mesh refinement (AMR) we store refinement trees in the cells of an unstructured coarse mesh. This lets us combine the speed and simpler management of structured refinement trees with the more flexible mesh generation of the unstructured coarse mesh. But this creates a conflict between performance and geometrical accuracy. If we favor speed we reduce the cells in our coarse mesh and hence reduce the accuracy of our geometrical representation. If we want more accurate results we generate a finer coarse mesh and lose performance by managing more cells in our unstructured coarse mesh. To mitigate this conflict we present the prototype of an geometry description which we implement in an already existing library. With this description we build geometry adapted hexahedral refinement trees, which also support high-order curved boundary cells. We also present examples on how to use this description. Moreover, we test the speedup of this new algorithm compared with coarse meshes with different geometrical errors.